How Do I Integrate d³k for Particle Number Calculations?

[sunipa.som]

I have to do integration
int_d^3 k
we know
v=(4/3)*pi*k^3
dv=4*pi*k^2 dk

can I write
int_d^3 k=int_dv=int_4*pi*k^2 dk ?
or I have to write
int_d^3 k=(1/2*pi)^3 *4*pi*k^2 dk

 

[HallsofIvy]

I'm afraid you will have to explain your notation. I assume that "int_d^3" refers to an integral in three dimensions but what is k?

v= (4/3) pi k^3 is the volume of a sphere of radius k. Is k what would normally be called $\rho$ ("rho") in spherical coordinates?

If so then your "int_d^3 k" would be
$$\int_{\theta= 0}^{2\pi}\int_{\phi= 0}^{\pi}\int_{\rho= 0}^R \rho (\rho^2 sin(\phi) d\theta d\phi d\rho)$$
$$= 2\pi\left(\int_{\phi=0}^\pi sin(\phi)d\phi\right)\left(\int_{\rho= 0}^R \rho^3 d\rho\right)$$
$$= 2\pi \left(2\right)\left(\frac{1}{4}\rho^4\right)_0^R= \pi R^4$$

 

[sunipa.som]

Actually I have number of particles N(r,k) as a function of r and k. r=0 to 50, k=0 to 50. I have to calculate total number of particles over whole volume.
So, i want to do integration
N_tot=int_int_N(r,k)*d^3 k*d^3 r
Now, int_d^3 k=int_4*pi*k^2 dk
int_d^3 r=int_4*pi*r^2 dr

N_tot=int_int_N(r,k)*d^3 k*d^3 r=int_int_N(r,k)*4*pi*k^2 dk*4*pi*r^2 dr

am I right?